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Equivalent norms in a Banach function space and the subsequence property
Jose M. Calabuig,Maite Fernandez-Unzueta,Fernando Galaz-Fontes,Enrique A. Sanchez-Perez 대한수학회 2019 대한수학회지 Vol.56 No.5
Consider a finite measure space $(\Ome,\Sig,\mu)$ and a Banach space $X(\mu)$ consisting of (equivalence classes of) real measurable functions defined on $\Ome$ such that $f\chi_A \in X(\mu) $ and $ \|f\chi_A \| \leq \|f\|, \ \pt f \in X(\mu), \ A \in \Sig$. We prove that if it satisfies the subsequence property, then it is an ideal of measurable functions and has an equivalent norm under which it is a Banach function space. As an application we characterize norms that are equivalent to a Banach function space norm.
EQUIVALENT NORMS IN A BANACH FUNCTION SPACE AND THE SUBSEQUENCE PROPERTY
Calabuig, Jose M.,Fernandez-Unzueta, Maite,Galaz-Fontes, Fernando,Sanchez-Perez, Enrique A. Korean Mathematical Society 2019 대한수학회지 Vol.56 No.5
Consider a finite measure space (${\Omega}$, ${\Sigma}$, ${\mu}$) and a Banach space $X({\mu})$ consisting of (equivalence classes of) real measurable functions defined on ${\Omega}$ such that $f{\chi}_A{\in}X({\mu})$ and ${\parallel}f{\chi}_A{\parallel}{\leq}{\parallel}f{\parallel}$, ${\forall}f{\in}({\mu})$, $A{\in}{\Sigma}$. We prove that if it satisfies the subsequence property, then it is an ideal of measurable functions and has an equivalent norm under which it is a Banach function space. As an application we characterize norms that are equivalent to a Banach function space norm.